On the probability of generating a minimal \(d\)-generated group (Q2765549)

If \(G\) is a finite group, let \(d(G)\) denote the minimal size of a generating set for \(G\) and \(\phi_G(s)\) denotes the number of \(s\)-tuples \((g_1,\dots,g_s)\in G^s\) which form a generating set for \(G\). The ratio \(P_G(s):=\phi_G(s)/|G|^s\) is the probability that a random \(s\)-tuple generates \(G\). This paper considers -- for a special class of groups -- the following conjecture of I. Pak (unpublished): for each real \(\alpha\) with \(0<\alpha<1\) there exists \(\beta>0\) such that for every finite group \(G\) we have \(P_G(s)\geq\alpha\) whenever \(s\geq\beta d(G)\log\log|G|\).NEWLINENEWLINENEWLINEIn an earlier paper [\textit{F. Dalla Volta} and \textit{A. Lucchini}, J. Aust. Math. Soc., Ser. A 64, No. 1, 82-91 (1998; Zbl 0902.20013)] two of the authors have introduced the class \(\mathcal L\) of finite groups. A group \(L\in{\mathcal L}\) if and only if it has a unique minimal normal subgroup, say \(M\), and \(M\) has a complement in \(L\) whenever \(M\) is Abelian. For each \(L\in{\mathcal L}\) and each integer \(t>0\), they defined NEWLINE\[NEWLINEL_t:=\{(l_1,\dots,l_t)\in L^t\mid l_1\equiv\cdots\equiv l_t\bmod M\}NEWLINE\]NEWLINE and showed that the class of all such \(L_t\) is precisely the class of groups \(G\) such that \(d(G)>d(G/N)\) for all nontrivial normal subgroups \(N\). Write \(\overline L_t:=L_t/\text{soc}(L_t)\cong L/M\).NEWLINENEWLINENEWLINEThe main theorem of the present paper is the following. For each \(\alpha\) with \(0<\alpha<1\) there exist positive \(\beta_1\) and \(\beta_2\) such that for each of the groups \(L_t\) of this form we have \(P_{L_t}(s)\geq\alpha P_{\overline L_t}(s)\) for all \(s\geq\beta_1+d(L_t)\) (if \(\text{soc}(L)\) is Abelian) or for all \(s\geq\beta_2\log(t+1)\) (if \(\text{soc}(L)\) is non-Abelian).